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$f: \{0, 1\}^* \to \mathbb{N}$, where $f(x) = k$ if the length of the string is $k$.

Can someone translate this for me, I do not quite understand what the star means at the end of the $\{0,1\}^*$.

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If $A$ is a set of symbols then $A^*$ is the set of all strings (words, sequences) built from elements of $A$. So $\{0,1\}^*$ is the set of all bit strings, like $1001$ and $10101000$.

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    $\begingroup$ or $001001110$ (just so OP doesn't get any wrong ideas) $\endgroup$ – SK19 May 8 '18 at 22:59

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